Download Wolfram Notebook

A geodesic is a locally arc length-minimizing curve. Equivalently, when parametrized by arc length, it has zero covariant derivative of its tangent vector. In the plane, the geodesics are straight lines. On the sphere, the geodesics are great circles (like the equator). The geodesics in a space depend on the Riemannian metric, which affects the notions of distance and acceleration.

Geodesics preserve a direction on a surface (Tietze 1965, pp. 26-27) and have many other interesting properties. At any point where its curvature is nonzero, the principal normal vector of a geodesic arc lies along the normal vector to the surface (Weinstock 1974, p. 65).

Every smooth Riemannian metric on the two-dimensional sphere has infinitely many geometrically distinct closed geodesics. This extends earlier results of Birkhoff and of Lyusternik and Schnirelmann establishing at least one and at least three closed geodesics, respectively, on such a sphere (Cipra 1993).

For a surface given parametrically by , , and , the geodesic can be found by minimizing the arc length

But

and similarly for and . Plugging in,

This can be rewritten as

where , , and

Starting with equation (6)

and taking derivatives,

so the Euler-Lagrange differential equation then gives

On a regular coordinate chart, , , and . In the special case when , , and are explicit functions of only, the conserved signed constant satisfies

For a finite slope , regularity implies . Only the branch satisfying the unsquared conservation equation is retained.

Now, if and are explicit functions of only and ,

where , and the sign is determined by the unsquared conservation equation. Hence

where is an arbitrary constant. In the case where and are explicit functions of only, then

so

and

where , in this conservation equation, and either sign corresponds to a local choice of direction. Constant solutions occur when and must be considered separately.

For a surface of revolution in which is rotated about the x-axis so that the equation of the surface is

the surface can be parameterized by

The equation of the geodesics is then

on a patch with .

See also

Blaschke Conjecture, Closed Geodesic, Conjugate Point, Ellipsoid Geodesic, Geodesic Curvature, Geodesic Dome, Geodesic Equation, Geodesic Mapping, Geodesic Metric Space, Geodesic Triangle, Graph Geodesic, Great Circle, Harmonic Map, Jacobi Field, Oblate Spheroid Geodesic, Paraboloid Geodesic, Wiedersehen Surface, Zoll Surface

Portions of this entry contributed by Todd Rowland

Explore with Wolfram|Alpha

References

Cipra, B. What's Happening in the Mathematical Sciences, Vol. 1. Providence, RI: Amer. Math. Soc., 1993.Tietze, H. Famous Problems of Mathematics: Solved and Unsolved Mathematics Problems from Antiquity to Modern Times. New York: Graylock Press, 1965.Weinstock, R. Calculus of Variations, with Applications to Physics and Engineering. New York: Dover, 1974.Weyl, H. Mathematische Analyse des Raumproblems: Vorlesungen, gehalten in Barcelona und Madrid. Berlin, Germany: Springer, 1923.

Referenced on Wolfram|Alpha

Geodesic

Cite this as:

Weisstein, Eric W., with contributions by Todd Rowland. "Geodesic." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Geodesic.html

Subject classifications